
Assume \({\cal X}\)  is represented in the TT format (REF ) with left-orthogonal core tensors \(\left\lbrace {\cal X}^1,\cdots ,{\cal X}^d\right\rbrace \) ,
i.e., \(L\left({\cal X}^k\right)^\mathsf {T}L\left({\cal X}^k\right)= \mathbf {I}_{r_k}\) , for \(k = 1,\cdots ,d-1\) . The tangent space of \({\cal M}_{\mathbf {r}}\)  at \({\cal X}\)  is given by [1]}
\(T_{{\cal X}} {\cal M}_{\mathbf {r}} = \left\lbrace \sum _{k = 1}^d\phi \left(\left\lbrace {\cal X}^1,\cdots ,\delta {\cal X}^k,\cdots ,{\cal X}^d\right\rbrace \right)\bigg \vert ~\delta {\cal X}^k\in \mathbb {R}^{r_{k-1}\times n_k\times r_k},L\left(\delta {\cal X}^k\right)^\mathsf {T}L\left({\cal X}^k\right)= \mathbf {0},k = 1,\cdots ,d-1\right\rbrace .\) 
